[math]\lim_{t\to-2}14\ln{\sqrt[4]{x+2}}\Biggr]_t^{14}[/math]...Which, as far as I know, should be divergent, since ln 0 is undefined, approaching negative infinity. However, (since this is online and I can attempt this problem multiple times) I know that it converges, but I don't understand how (unless I am right and webassign is wrong).

I actually do know the answer, which is 448/3 (which I arrived at with a calculator and the ability to try a problem almost exactly the same, whose answer I could see after not getting it right--not the most enlightening process), but I don't understand how it is 448/3. So, if anybody understand why it is that...please help me see why.

skullturf wrote:I often tell my students to be careful with that casual paraphrase. Notice that, for example,[math]\int \frac1{x^2+1} dx = \arctan x[/math]which has absolutely nothing to do with logarithms.

Well, not nothing. 1/(x2+1) = i/2 (1/(i+x) + 1/(i-x)). So its integral is i/2 (log(i+x) - log(i-x)) = i/2 log((i+x)/(i-x)). You can see that this is the same thing as arctan x, since when x is real it is the imaginary part of log(1+ix).

Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

skullturf wrote:I teach calculus. Both you and my students know the formula[math]\int \frac1x dx = \ln x[/math]which, casually paraphrased, could be said as "integral of 1 over something is natural log of something."

This is why I beat people over the head for saying those kinds of things, even if it's a casual paraphrase.I find it better to say "integral of 1 over x plus something is is natural log of x plus something".

It's not much harder to say, but you won't get tripped up on something like 1 / (x^2 + 1).

http://en.wikipedia.org/wiki/DSV_Alvin#Sinking wrote:Researchers found a cheese sandwich which exhibited no visible signs of decomposition, and was in fact eaten.

skullturf wrote:I teach calculus. Both you and my students know the formula[math]\int \frac1x dx = \ln x[/math]which, casually paraphrased, could be said as "integral of 1 over something is natural log of something."

I often tell my students to be careful with that casual paraphrase. Notice that, for example,[math]\int \frac1{x^2+1} dx = \arctan x[/math]which has absolutely nothing to do with logarithms.